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The efficient numerical solution of PDE constrained optimization problems plays an important role in many engineering and science applications. The development of robust and efficient numerical algorithms requires the integration of tools from several mathematical subdisciplines, often only described individually in books or journal articles. The goal of this book is to provide readers with a brief introduction to this active research area as well as with an overview of the state-of-the-art in the important topics of adaptive discretizations of PDE optimization problems, handling of control and state constraints, domain decomposition and homogenization of PDEs on networks, and reduced order modeling.